Magnetophoresis is analogous to dielectrophoresis, but leads to vastly different experimental realizations owing to
(1) the enormous variation in magnetic permeability observed between different materials, (2) the
nonlinearity of magnetic response of most materials to common experimental magnetic fields, and (3) the
different experimental tools available to generate magnetic fields. Magnetic fields are an excellent way
to apply forces to a controlled subset of the particles or cells or analytes in a system, because only
only specialized materials are affected by magnetic fields enough to be successfully actuated in most
microsystems.

17.2.1 Origin of magnetic fields in materials

Classically, magnetic fields in materials can be thought of as being induced by orbital rotation of electrons,
analogous to the magnetic field generated by current flowing through a wire loop. This classical representation can
qualitatively explain differences between different material classes (e.g., diamagnetic, paramagnetic, ferromagnetic)
but cannot predict how these properties stem from molecular structure—that requires a quantum-mechanical
description.

Diamagnetism
Diamagnetic materials are those whose electrons are all paired. Diamagnetic effects stem from changes in
electron orbital motion induced by a magnetic field. It leads to a dipole aligned against the magnetic field, i.e., the
diamagnetic component of magnetic permeability is negative. Diamagnetic effects are small enough to be neglected
for our purposes.

Paramagnetism
Paramagnetic effects occur in materials whose electrons are unpaired. In this case, the spins of
the unpaired electrons (and the resulting magnetic moment) align with the external magnetic field.
Paramagnetism is exhibited when thermal fluctuations prevent the magnetic dipoles from locking in orientation
aligned with the field. In this case, the paramagnetic component of magnetic permeability is positive but
small.

Ferromagnetism
Ferromagnetic effects occur in materials whose electrons are unpaired. In this case, the spins of the unpaired
electrons (and the resulting magnetic moment) aligns with the external magnetic field. Ferromagnetism is exhibited
when thermal fluctuations are small as compared to the forces that lead magnetic dipoles to lock in orientation
aligned with the field. In this case, the ferromagnetic component of magnetic permeability is positive and large. The
only difference between a magnet and the material to which magnets are attracted (e.g., iron) is the size of the
magnetic domains. All ferromagnetic materials have magnetic domains in which the magnetic dipoles
are aligned, but in materials such as iron, the magnetic domains are small and randomly oriented,
leading to no net orientation on a large scale. A magnet, in contrast, has large, permanently oriented
domains.

17.2.2 Attributes of magnetism

While magnetism is closely related to electricity, and the equations are similar, key differences between the two
make the engineering application of magnetic effects different from application of electrical effects. Electricity is
controlled primarily bymonopoles (point charges, for example an ion) and dipoles (equal and opposite charges that
are separated by a distance, for example a nucleus and an electron cloud that has been polarized by an electric field).
Magnetism has no monopoles and is thus controlled primarily by dipole or polarization effects. Thus electricity is
characterized by both conductivity and electrical permittivity, while magnetism is described only bymagnetic
permeability. The magnetic permeabilities of most materials are essentially the same as free space, but the
magnetic permeability of a few select materials (such as iron, nickel, ferrite, etc.) can be 3-6 orders of
magnitude higher. This can be seen quantitatively in Table 17.1. In contrast, electric permittivitiesvary
continuously among materials and by not more than a factor of 80 or so from the lowest (air; 1) to the highest
(water; 80). The impact of this is that (a) most materials do not experience magnetic forces that are
significant, and (b) the materials that do experience magnetic forces are specialized and relatively
easy to engineer. In a microsystem, magnetic fields can manipulatemagnetic beads without affecting
other attributes of the flow. This, in general, can not be said for electrical effects likeelectrophoresis
ordielectrophoresis. Because of this, magnetic beads can be used to isolate forces in specific areas or to specific
particles.

The magnetic permeability μmag expresses the magnetic behavior of a material:

(17.53)

where is the appliedmagnetic field and is theinduced magnetic field. is the resultingmagnetization of the
material. Themagnetic permeability of free space μmag,0 is 4π×10-7H∕m. The magnetic permeability of other
materials are given by μmag= μmag,0(1+χm), where χm is themagnetic susceptibility of the material. Most
materials have values of χm that are small as compared to unity, and thus magnetic effects are tiny. Only
ferro-, antiferro- and ferrimagnetic materials experience significant magnetic effects in systems of
interest.

Material Class

Examples

Typical χm

B-H relationship

Comments

diamagnetic

water

-l×10-5

linear (constant χm)

no hysteresis

paramagnetic

aluminum

2×10-5

linear (constant χm)

no hysteresis; becomes
ferromagnetic below
Curie temp

ferromagnetic

Iron

3×103

nonlinear (χm is f(B))

shows hysteresis

ferrimagnetic

MnZn(Fe204)2

2.5×10-3

nonlinear (χm is f(B))

shows hysteresis

Table 17.1: Properties of magnetic materials.

Saturation

Saturationis a key property that prevents magnetophoresis from being directly analogous to dielectrophoresis.
Materials have a saturation magnetism (e.g., 2.2T for iron) which corresponds to their spins being fully aligned.
Increased magnetic field does not lead to further spin or further magnetic flux density. This leads to a magnetic
permeability that is dependent on the applied magnetic field (See Figure 17.13a). In magnetically soft materials,
the B-H curve is nonlinear but shows no hysteresis. In magnetically hard materials, the B-H curve
also shows hysteresis (Figure 17.13b), owing to the fact that nonequilibrium states are metastable.

Figure 17.13: (a):Saturation in a magnetically soft material with no hysteresis. This is representative
ofsuperparamagnetism, the property magnetic beads are designed to exhibit. Superparamagnetism is
typically observed in small domains of ferromagnetic materials. (b): saturation in a magnetically hard
material with hysteresis, such as iron or magnetite at long length scales. Mrem is the remanence magnetism.
Hc is the coercivity. Reproduced from [38].

17.2.3 Magnetic properties of superparamagnetic beads

Superparamagnetic beads typically consist of a polystyrene matrix (on order 1 μm diameter) filled to about 15%
mass fraction with roughly 10 nm ferro- or ferrimagnetic particles. The 10 nm particles are superparamagnetic,
which means that they respond strongly to external magnetic fields, but do not retain permanent magnetism, and do
not showhysteresis. Magnetization is inherently unstable, because the orientation of magnetic dipoles can induce
other dipoles to also align. In diamagnetic materials, all electrons are paired and this effect does not
occur. Paramagnetic and ferromagnetic materials have unpaired electrons and magnetic dipoles can
become locked. Above theCurie temperature, these materials do not show magnetic dipole locking
because thermal fluctuations dominate; they are termed paramagnetic and magnetic effects are weak.
Below the Curie temperature, the dipoles tend to lock, making the induced magnetic field H a nonlinear
function of the applied magnetic field B, and also leading to hysteresis and relaxation effects, since locked
dipoles equilibrate slowly when fields are removed. Superparamagnetic particles are small ferro- or
ferrimagnetic particles which, owing to their small size, can orient and equilibrate themselves quickly, so they
show strong effects like a ferro/ferrimagnetic material, while also avoiding hysteresis and permanent
magnetism.

17.2.4 Magnetophoretic forces

Magnetophoresis is the motion of objects with respect to a surrounding medium caused by the net interaction of a
magnetization with an magnetic field gradient. Magnetophoresis is analogous to dielectrophoresis of a particle
except that magnetic materials have a nonlinear induced dipole response when exposed to typical experimental
conditions.

The magnetophoretic force on a particle mag is given by

(17.54)

where is the external magnetic field, μmag,m is the magnetic permeability of the medium, and meff is
the magnetic dipole moment (i.e., magnetization) induced by the field. This equation is analogous to
Equation 17.26—the inclusion of themagnetic permeability here (when no electric permittivity was used in
Equation 17.26) is strictly attributable to conventions used when defining magnetic moments. The force relations
for DEP and magnetophoresis are analogous.

17.2.5 DC magnetophoresis of spheres—linear limit

If we assume a steady magnetic field and particles that respond linearly to the magnetic field, then magnetophoresis
is quite similar to DC dielectrophoresis. For spheres of diameter a, we can show that the averaged magnetophoretic
force mag is given by

Here, as before, subscripts m and p indicate medium and particle, respectively. Here theClausius-Mossotti factor
uses magnetic permeabilities rather than electrical permittivities, as was the case for DEP. Because we are
considering only DC magnetic fields, the magnetic Clausius-Mossotti factor is real. The prefactor of 2 also stems
from use of a DC magnetic field. Commercial particles with roughly 10-15% ferrite or magnetite have
effectivemagnetic susceptibilities χm on the order of 3 and a roughly linear range up to applied magnetic fields of
approximately H = 5×104A∕m. Owing to the nonlinearities in magnetic systems, calculations using
Equation 17.55 are approximate.